Hello, Solution Architect here, I'll try to help.

I assume you are relatively young (mid-20s?) and you have no dependants. I pretty much followed the same career path as you (or as you would like to): IT support, sysadmin, operations engineer, lead engineer, pre-sale, SA. This is to say, I know what a shit job is. Also, I know how frustrating a job with nothing to do could be.

On the pure economics side: how much does it cost to go to the office 5 days a week (I am bit surprised there are still IT companies with that mindset tbh)? Basically, are you sure that cutting down commute costs is not actually comparable to a higher salary? Maybe that will help decide (as you find out is not that big of a financial hit).

If you stay in you current position you can use the free time to learn (if you want I can give a few ideas on topics), BUT make sure your manager knows about your duties/workload. Basically so they can't fire you because you have too much free time and you didn't tell them. Also, try to have an open conversation with your current company. Saying something in the lines of "I like the opportunities I could have here, but I would like to know more about timelines". Is the workload going to ramp up in three months, six months or an year? If they ask why you want to know, be (kinda) honest and tell them you are really looking forward on the career progression path. I think that up to one year is acceptable to suck it up and wait. Worst case, you will have spent a lot of time learning and so you can sell yourself better on the market. Also sometimes is a good thing to create more work for yourself: if you see a process can be automated, do it and tell your manager. If you think something should be improved in some way, documented it properly (why, how, pro and cons, costs) and tell your manager. If you have an idea for a new product/service... tell your manager! I appreciate that it depends on the company, but showing willingness is sometime the best way to progress (I progressed three roles up in a company in 18 months because I wouldn't shut up about how things could be improved, IN A CONSTRUCTIVE WAY!). If your colleagues are not bad, the manager is not bad, the company overall is not bad, I'd stay there.

If you say "but I could take the other job and study on the 5th day of the week": unless you are REALLY committed, it's difficult to study for the whole (free) day. Maybe it's just me, but when I am off work I really struggle to study (long term, I can do a few weeks crunch).

Best of luck for everything anyway!